Let $ABCD$ be a rectangle such that $AB=5$ and $BC=12$. There exist two distinct points $X_1$ and $X_2$ on $BC$ such that $\angle AX_1D=\angle AX_2D=90^\circ$. Suppose that $d$ is the distance from $X_1$ to $X_2$. What is $d^2$?

This problem is posed by David A.